If your trademark is not fully registered yet, you have limited rights.
Let's talk about other applications first. Once you file your application, you gain legal grounds over anyone who applies with an identical or "confusingly similar" name after you. In some jurisdictions, the examiner will temporarily stop such applications based on the likelihood of confusion.
However, if you find out a 3rd party is copying your brand, you should wait until you obtain trademark registration. You can then send a cease and desist letter to stop the 3rd party from using your name, as this would be considered trademark infringement.
In some jurisdictions, including the EU and the UK, your trademark rights are retroactive to the date of filing. In simple terms, if someone copies your brand or uses a confusingly similar name during the period when your trademark is pending, you can take legal action once your trademark is fully registered.